Theorem ltexnq 10662

Description: Ordering on positive fractions in terms of existence of sum. Definition in Proposition 9-2.6 of [Gleason] p. 119. (Contributed by NM, 24-Apr-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
ltexnq	⊢ (𝐵 ∈ Q → (𝐴 <Q 𝐵 ↔ ∃𝑥(𝐴 +Q 𝑥) = 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem ltexnq

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltrelnq 10613	. . . 4 ⊢ <Q ⊆ (Q × Q)
2	1	brel 5643	. . 3 ⊢ (𝐴 <Q 𝐵 → (𝐴 ∈ Q ∧ 𝐵 ∈ Q))
3		ordpinq 10630	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴))))
4		elpqn 10612	. . . . . . . . 9 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
5	4	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 𝐴 ∈ (N × N))
6		xp1st 7836	. . . . . . . 8 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N)
7	5, 6	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (1st ‘𝐴) ∈ N)
8		elpqn 10612	. . . . . . . . 9 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N))
9	8	adantl 481	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 𝐵 ∈ (N × N))
10		xp2nd 7837	. . . . . . . 8 ⊢ (𝐵 ∈ (N × N) → (2nd ‘𝐵) ∈ N)
11	9, 10	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (2nd ‘𝐵) ∈ N)
12		mulclpi 10580	. . . . . . 7 ⊢ (((1st ‘𝐴) ∈ N ∧ (2nd ‘𝐵) ∈ N) → ((1st ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
13	7, 11, 12	syl2anc 583	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((1st ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
14		xp1st 7836	. . . . . . . 8 ⊢ (𝐵 ∈ (N × N) → (1st ‘𝐵) ∈ N)
15	9, 14	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (1st ‘𝐵) ∈ N)
16		xp2nd 7837	. . . . . . . 8 ⊢ (𝐴 ∈ (N × N) → (2nd ‘𝐴) ∈ N)
17	5, 16	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (2nd ‘𝐴) ∈ N)
18		mulclpi 10580	. . . . . . 7 ⊢ (((1st ‘𝐵) ∈ N ∧ (2nd ‘𝐴) ∈ N) → ((1st ‘𝐵) ·N (2nd ‘𝐴)) ∈ N)
19	15, 17, 18	syl2anc 583	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((1st ‘𝐵) ·N (2nd ‘𝐴)) ∈ N)
20		ltexpi 10589	. . . . . 6 ⊢ ((((1st ‘𝐴) ·N (2nd ‘𝐵)) ∈ N ∧ ((1st ‘𝐵) ·N (2nd ‘𝐴)) ∈ N) → (((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴)) ↔ ∃𝑦 ∈ N (((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴))))
21	13, 19, 20	syl2anc 583	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴)) ↔ ∃𝑦 ∈ N (((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴))))
22		relxp 5598	. . . . . . . . . . . 12 ⊢ Rel (N × N)
23	4	ad2antrr 722	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → 𝐴 ∈ (N × N))
24		1st2nd 7853	. . . . . . . . . . . 12 ⊢ ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
25	22, 23, 24	sylancr 586	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
26	25	oveq1d 7270	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩))
27	7	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (1st ‘𝐴) ∈ N)
28	17	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (2nd ‘𝐴) ∈ N)
29		simpr 484	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → 𝑦 ∈ N)
30		mulclpi 10580	. . . . . . . . . . . . 13 ⊢ (((2nd ‘𝐴) ∈ N ∧ (2nd ‘𝐵) ∈ N) → ((2nd ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
31	17, 11, 30	syl2anc 583	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((2nd ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
32	31	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ((2nd ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
33		addpipq 10624	. . . . . . . . . . 11 ⊢ ((((1st ‘𝐴) ∈ N ∧ (2nd ‘𝐴) ∈ N) ∧ (𝑦 ∈ N ∧ ((2nd ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)) → (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = ⟨(((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) +N (𝑦 ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))⟩)
34	27, 28, 29, 32, 33	syl22anc 835	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = ⟨(((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) +N (𝑦 ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))⟩)
35	26, 34	eqtrd 2778	. . . . . . . . 9 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = ⟨(((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) +N (𝑦 ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))⟩)
36		oveq2 7263	. . . . . . . . . . . 12 ⊢ ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → ((2nd ‘𝐴) ·N (((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦)) = ((2nd ‘𝐴) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴))))
37		distrpi 10585	. . . . . . . . . . . . 13 ⊢ ((2nd ‘𝐴) ·N (((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦)) = (((2nd ‘𝐴) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) +N ((2nd ‘𝐴) ·N 𝑦))
38		fvex 6769	. . . . . . . . . . . . . . 15 ⊢ (2nd ‘𝐴) ∈ V
39		fvex 6769	. . . . . . . . . . . . . . 15 ⊢ (1st ‘𝐴) ∈ V
40		fvex 6769	. . . . . . . . . . . . . . 15 ⊢ (2nd ‘𝐵) ∈ V
41		mulcompi 10583	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
42		mulasspi 10584	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
43	38, 39, 40, 41, 42	caov12 7478	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝐴) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) = ((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))
44		mulcompi 10583	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝐴) ·N 𝑦) = (𝑦 ·N (2nd ‘𝐴))
45	43, 44	oveq12i 7267	. . . . . . . . . . . . 13 ⊢ (((2nd ‘𝐴) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) +N ((2nd ‘𝐴) ·N 𝑦)) = (((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) +N (𝑦 ·N (2nd ‘𝐴)))
46	37, 45	eqtr2i 2767	. . . . . . . . . . . 12 ⊢ (((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) +N (𝑦 ·N (2nd ‘𝐴))) = ((2nd ‘𝐴) ·N (((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦))
47		mulasspi 10584	. . . . . . . . . . . . 13 ⊢ (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)) = ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (1st ‘𝐵)))
48		mulcompi 10583	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝐴) ·N (1st ‘𝐵)) = ((1st ‘𝐵) ·N (2nd ‘𝐴))
49	48	oveq2i 7266	. . . . . . . . . . . . 13 ⊢ ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (1st ‘𝐵))) = ((2nd ‘𝐴) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴)))
50	47, 49	eqtri 2766	. . . . . . . . . . . 12 ⊢ (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)) = ((2nd ‘𝐴) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴)))
51	36, 46, 50	3eqtr4g 2804	. . . . . . . . . . 11 ⊢ ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → (((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) +N (𝑦 ·N (2nd ‘𝐴))) = (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)))
52		mulasspi 10584	. . . . . . . . . . . . 13 ⊢ (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵)) = ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))
53	52	eqcomi 2747	. . . . . . . . . . . 12 ⊢ ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) = (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))
54	53	a1i 11	. . . . . . . . . . 11 ⊢ ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) = (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵)))
55	51, 54	opeq12d 4809	. . . . . . . . . 10 ⊢ ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → ⟨(((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) +N (𝑦 ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))⟩ = ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩)
56	55	eqeq2d 2749	. . . . . . . . 9 ⊢ ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → ((𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = ⟨(((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) +N (𝑦 ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))⟩ ↔ (𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩))
57	35, 56	syl5ibcom 244	. . . . . . . 8 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → (𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩))
58		fveq2 6756	. . . . . . . . 9 ⊢ ((𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ → ([Q]‘(𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = ([Q]‘⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩))
59		adderpq 10643	. . . . . . . . . . 11 ⊢ (([Q]‘𝐴) +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = ([Q]‘(𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩))
60		nqerid 10620	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Q → ([Q]‘𝐴) = 𝐴)
61	60	ad2antrr 722	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ([Q]‘𝐴) = 𝐴)
62	61	oveq1d 7270	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (([Q]‘𝐴) +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = (𝐴 +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)))
63	59, 62	eqtr3id 2793	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ([Q]‘(𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = (𝐴 +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)))
64		mulclpi 10580	. . . . . . . . . . . . . . . 16 ⊢ (((2nd ‘𝐴) ∈ N ∧ (2nd ‘𝐴) ∈ N) → ((2nd ‘𝐴) ·N (2nd ‘𝐴)) ∈ N)
65	17, 17, 64	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((2nd ‘𝐴) ·N (2nd ‘𝐴)) ∈ N)
66	65	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ((2nd ‘𝐴) ·N (2nd ‘𝐴)) ∈ N)
67	15	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (1st ‘𝐵) ∈ N)
68	11	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (2nd ‘𝐵) ∈ N)
69		mulcanenq 10647	. . . . . . . . . . . . . 14 ⊢ ((((2nd ‘𝐴) ·N (2nd ‘𝐴)) ∈ N ∧ (1st ‘𝐵) ∈ N ∧ (2nd ‘𝐵) ∈ N) → ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ ~Q ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩)
70	66, 67, 68, 69	syl3anc 1369	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ ~Q ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩)
71	8	ad2antlr 723	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → 𝐵 ∈ (N × N))
72		1st2nd 7853	. . . . . . . . . . . . . 14 ⊢ ((Rel (N × N) ∧ 𝐵 ∈ (N × N)) → 𝐵 = ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩)
73	22, 71, 72	sylancr 586	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → 𝐵 = ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩)
74	70, 73	breqtrrd 5098	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ ~Q 𝐵)
75		mulclpi 10580	. . . . . . . . . . . . . . 15 ⊢ ((((2nd ‘𝐴) ·N (2nd ‘𝐴)) ∈ N ∧ (1st ‘𝐵) ∈ N) → (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)) ∈ N)
76	66, 67, 75	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)) ∈ N)
77		mulclpi 10580	. . . . . . . . . . . . . . 15 ⊢ ((((2nd ‘𝐴) ·N (2nd ‘𝐴)) ∈ N ∧ (2nd ‘𝐵) ∈ N) → (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵)) ∈ N)
78	66, 68, 77	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵)) ∈ N)
79	76, 78	opelxpd 5618	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ ∈ (N × N))
80		nqereq 10622	. . . . . . . . . . . . 13 ⊢ ((⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ ~Q 𝐵 ↔ ([Q]‘⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩) = ([Q]‘𝐵)))
81	79, 71, 80	syl2anc 583	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ ~Q 𝐵 ↔ ([Q]‘⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩) = ([Q]‘𝐵)))
82	74, 81	mpbid 231	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ([Q]‘⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩) = ([Q]‘𝐵))
83		nqerid 10620	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ Q → ([Q]‘𝐵) = 𝐵)
84	83	ad2antlr 723	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ([Q]‘𝐵) = 𝐵)
85	82, 84	eqtrd 2778	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ([Q]‘⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩) = 𝐵)
86	63, 85	eqeq12d 2754	. . . . . . . . 9 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (([Q]‘(𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = ([Q]‘⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩) ↔ (𝐴 +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = 𝐵))
87	58, 86	syl5ib 243	. . . . . . . 8 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ((𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ → (𝐴 +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = 𝐵))
88	57, 87	syld 47	. . . . . . 7 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → (𝐴 +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = 𝐵))
89		fvex 6769	. . . . . . . 8 ⊢ ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) ∈ V
90		oveq2 7263	. . . . . . . . 9 ⊢ (𝑥 = ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) → (𝐴 +Q 𝑥) = (𝐴 +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)))
91	90	eqeq1d 2740	. . . . . . . 8 ⊢ (𝑥 = ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) → ((𝐴 +Q 𝑥) = 𝐵 ↔ (𝐴 +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = 𝐵))
92	89, 91	spcev 3535	. . . . . . 7 ⊢ ((𝐴 +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = 𝐵 → ∃𝑥(𝐴 +Q 𝑥) = 𝐵)
93	88, 92	syl6 35	. . . . . 6 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → ∃𝑥(𝐴 +Q 𝑥) = 𝐵))
94	93	rexlimdva 3212	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (∃𝑦 ∈ N (((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → ∃𝑥(𝐴 +Q 𝑥) = 𝐵))
95	21, 94	sylbid 239	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴)) → ∃𝑥(𝐴 +Q 𝑥) = 𝐵))
96	3, 95	sylbid 239	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 → ∃𝑥(𝐴 +Q 𝑥) = 𝐵))
97	2, 96	mpcom 38	. 2 ⊢ (𝐴 <Q 𝐵 → ∃𝑥(𝐴 +Q 𝑥) = 𝐵)
98		eleq1 2826	. . . . . . 7 ⊢ ((𝐴 +Q 𝑥) = 𝐵 → ((𝐴 +Q 𝑥) ∈ Q ↔ 𝐵 ∈ Q))
99	98	biimparc 479	. . . . . 6 ⊢ ((𝐵 ∈ Q ∧ (𝐴 +Q 𝑥) = 𝐵) → (𝐴 +Q 𝑥) ∈ Q)
100		addnqf 10635	. . . . . . . 8 ⊢ +Q :(Q × Q)⟶Q
101	100	fdmi 6596	. . . . . . 7 ⊢ dom +Q = (Q × Q)
102		0nnq 10611	. . . . . . 7 ⊢ ¬ ∅ ∈ Q
103	101, 102	ndmovrcl 7436	. . . . . 6 ⊢ ((𝐴 +Q 𝑥) ∈ Q → (𝐴 ∈ Q ∧ 𝑥 ∈ Q))
104		ltaddnq 10661	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝑥 ∈ Q) → 𝐴 <Q (𝐴 +Q 𝑥))
105	99, 103, 104	3syl 18	. . . . 5 ⊢ ((𝐵 ∈ Q ∧ (𝐴 +Q 𝑥) = 𝐵) → 𝐴 <Q (𝐴 +Q 𝑥))
106		simpr 484	. . . . 5 ⊢ ((𝐵 ∈ Q ∧ (𝐴 +Q 𝑥) = 𝐵) → (𝐴 +Q 𝑥) = 𝐵)
107	105, 106	breqtrd 5096	. . . 4 ⊢ ((𝐵 ∈ Q ∧ (𝐴 +Q 𝑥) = 𝐵) → 𝐴 <Q 𝐵)
108	107	ex 412	. . 3 ⊢ (𝐵 ∈ Q → ((𝐴 +Q 𝑥) = 𝐵 → 𝐴 <Q 𝐵))
109	108	exlimdv 1937	. 2 ⊢ (𝐵 ∈ Q → (∃𝑥(𝐴 +Q 𝑥) = 𝐵 → 𝐴 <Q 𝐵))
110	97, 109	impbid2 225	1 ⊢ (𝐵 ∈ Q → (𝐴 <Q 𝐵 ↔ ∃𝑥(𝐴 +Q 𝑥) = 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  ∃wrex 3064  ⟨cop 4564   class class class wbr 5070   × cxp 5578  Rel wrel 5585  ‘cfv 6418  (class class class)co 7255  1^st c1st 7802  2^nd c2nd 7803  Ncnpi 10531   +_N cpli 10532   ·_N cmi 10533   <_N clti 10534   +_pQ cplpq 10535   ~_Q ceq 10538  Qcnq 10539  [Q]cerq 10541   +_Q cplq 10542   <_Q cltq 10545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-oadd 8271  df-omul 8272  df-er 8456  df-ni 10559  df-pli 10560  df-mi 10561  df-lti 10562  df-plpq 10595  df-mpq 10596  df-ltpq 10597  df-enq 10598  df-nq 10599  df-erq 10600  df-plq 10601  df-mq 10602  df-1nq 10603  df-ltnq 10605

This theorem is referenced by:  ltbtwnnq  10665  prnmadd  10684  ltexprlem4  10726  ltexprlem7  10729  prlem936  10734